Generalised Weyl theorems and spectral pollution in the Galerkin method

Lyonell Boulton (1) , Nabile Boussaid (2) , Mathieu Lewin (3)

1 Department of Mathematics
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We consider a general framework for investigating spectral pollution in the Galerkin method. We show how this phenomenon is characterised via the existence of particular Weyl sequences which are singular in a suitable sense. For a semi-bounded selfadjoint operator A we identify relative compactness conditions on a selfadjoint perturbation B ensuring that the limiting set of spectral pollution of A and B coincide. Our results show that, under perturbation, this limiting set behaves in a similar fashion as the essential spectrum.

Keywords: Weyl's Theorem; generalised essential spectrum; spectral pollution; Galerkin method

Domains: Spectral Theory [math.SP]

Lyonell Boulton, Nabile Boussaid, Mathieu Lewin. Generalised Weyl theorems and spectral pollution in the Galerkin method. Journal of Spectral Theory, 2012, 2 (4), pp.329-354. ⟨10.4171/JST/32⟩. ⟨hal-00536270v2⟩